441 research outputs found
Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications
A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases).
To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method.
To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively.
Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices.
Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective
Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications
A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases).
To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method.
To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively.
Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices.
Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective
Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution of the elliptic eigenvalue problem for the 3D
lattice-structured systems. We consider the linearized Hartree-Fock equation
over a spatial lattice for both periodic and
non-periodic problem setting, discretized in the localized Gaussian-type
orbitals basis. In the periodic case, the Galerkin system matrix obeys a
three-level block-circulant structure that allows the FFT-based
diagonalization, while for the finite extended systems in a box (Dirichlet
boundary conditions) we arrive at the perturbed block-Toeplitz representation
providing fast matrix-vector multiplication and low storage size. The proposed
grid-based tensor techniques manifest the twofold benefits: (a) the entries of
the Fock matrix are computed by 1D operations using low-rank tensors
represented on a 3D grid, (b) in the periodic case the low-rank tensor
structure in the diagonal blocks of the Fock matrix in the Fourier space
reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems
in a box with Dirichlet boundary conditions are treated numerically by our
previous tensor solver for single molecules, which makes possible calculations
on rather large lattices due to reduced numerical
cost for 3D problems. The numerical simulations for both box-type and periodic
lattice chain in a 3D rectangular "tube" with up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol
This note is devoted to preconditioning strategies for non-Hermitian
multilevel block Toeplitz linear systems associated with a multivariate
Lebesgue integrable matrix-valued symbol. In particular, we consider special
preconditioned matrices, where the preconditioner has a band multilevel block
Toeplitz structure, and we complement known results on the localization of the
spectrum with global distribution results for the eigenvalues of the
preconditioned matrices. In this respect, our main result is as follows. Let
, let be the linear space of complex matrices, and let be functions whose components
belong to .
Consider the matrices , where varies
in and are the multilevel block Toeplitz matrices
of size generated by . Then
, i.e. the family
of matrices has a global (asymptotic)
spectral distribution described by the function , provided
possesses certain properties (which ensure in particular the invertibility of
for all ) and the following topological conditions are met:
the essential range of , defined as the union of the essential ranges
of the eigenvalue functions , does not
disconnect the complex plane and has empty interior. This result generalizes
the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work,
concerning the non-preconditioned case . The last part of this note is
devoted to numerical experiments, which confirm the theoretical analysis and
suggest the choice of optimal GMRES preconditioning techniques to be used for
the considered linear systems.Comment: 18 pages, 26 figure
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